Subgroups — #13171

Order 256.

137 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#368	C₂	1	1	yes	maximal, characteristic
2	C₂×#366	C₂	1	1	yes	maximal, characteristic
3	#2108	C₂	1	1	yes	maximal, characteristic
4	#1952	C₂	1	2	yes	maximal
5	#1387	C₂	1	2	yes	maximal
6	C₂×M₃₂	C₂²	1	1	yes	characteristic
7	C₂²×C₁₆	C₂²	1	1	yes	characteristic
8	C₂×#108	C₂²	1	2	yes	—
9	#428	—	2	2	no	—
10	C₂×#107	C₂²	1	2	yes	—
11	#405	C₂²	1	1	yes	characteristic
12	#368	C₄	1	4	yes	complemented
13	#366	—	2	1	no	—
14	#366	—	2	1	no	—
15	#346	—	2	1	no	—
16	C₄×C₁₆	—	2	1	no	—
17	#343	—	2	2	no	—
18	C₂×M₁₆	—	2	2	no	—
19	C₂×M₁₆	D₄	1	1	yes	characteristic
20	C₂²×C₈	C₂³	1	1	yes	characteristic
21	#117	D₄	1	1	yes	characteristic
22	C₂×C₄⋊C₄	—	2	2	no	—
23	M₃₂	—	2	1	no	—
24	M₃₂	Q₈	1	2	yes	—
25	C₂×C₁₆	—	2	1	no	—
26	C₂×C₁₆	C₂×C₄	1	2	yes	—
27	C₂×C₁₆	D₄	1	1	yes	characteristic
28	C₂×C₁₆	D₄	1	1	yes	characteristic
29	C₂×C₁₆	Q₈	1	2	yes	—
30	#108	—	2	2	no	—
31	#108	C₂×C₄	1	4	yes	—
32	#107	—	2	2	no	—
33	#107	—	2	2	no	—
34	#107	—	4	2	no	—
35	#106	—	4	2	no	—
36	#97	—	2	1	no	—
37	C₄×C₈	—	2	1	no	—
38	C₂²×C₄	—	4	2	no	—
39	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
40	M₁₆	—	2	1	no	—
41	M₁₆	—	2	4	no	—
42	M₁₆	—	4	2	no	—
43	M₁₆	Q₁₆	1	2	yes	—
44	C₂×C₈	—	2	1	no	—
45	C₂×C₈	—	4	2	no	—
46	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₈	C₂×D₄	1	1	yes	characteristic
48	C₂×C₈	C₂×Q₈	1	1	yes	characteristic
49	C₂×C₈	C₂×Q₈	1	1	yes	characteristic
50	C₂×C₈	#41	1	1	yes	characteristic
51	C₂×C₈	#41	1	1	yes	characteristic
52	C₄⋊C₄	—	4	2	no	—
53	C₄⋊C₄	—	4	2	no	—
54	C₄⋊C₄	D₈	1	1	yes	characteristic
55	C₄⋊C₄	D₈	1	1	yes	characteristic
56	C₂²⋊C₄	—	2	1	no	—
57	C₄²	—	2	1	no	—
58	C₁₆	—	2	2	no	—
59	C₁₆	C₄⋊C₄	1	2	yes	—
60	C₁₆	C₄⋊C₄	1	2	yes	—
61	C₂³	#135	1	1	yes	characteristic
62	C₂×C₄	—	2	1	no	—
63	C₂×C₄	—	2	1	no	—
64	C₂×C₄	—	8	2	no	—
65	C₂×C₄	—	16	2	no	—
66	C₂×C₄	C₄×D₄	1	1	yes	characteristic
67	C₂×C₄	#121	1	1	yes	characteristic
68	C₂×C₄	C₄⋊Q₈	1	1	yes	characteristic
69	C₂×C₄	C₂×D₈	1	1	yes	characteristic
70	C₂×C₄	C₂×Q₁₆	1	1	yes	characteristic
71	C₂×C₄	#135	1	1	yes	characteristic
72	C₈	—	2	2	no	—
73	C₈	—	4	4	no	—
74	C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
75	C₈	C₄×Q₈	1	1	yes	characteristic
76	C₈	C₂²⋊Q₈	1	1	yes	characteristic
77	C₈	#125	1	1	yes	characteristic
78	C₂²	—	2	1	no	—
79	C₂²	—	2	1	no	—
80	C₂²	C₄×Q₁₆	1	1	yes	characteristic
81	C₂²	#466	1	1	yes	characteristic
82	C₂²	#499	1	1	yes	characteristic
83	C₄	—	4	1	no	—
84	C₄	—	4	1	no	—
85	C₄	—	16	2	no	—
86	C₄	#389	1	1	yes	characteristic
87	C₄	C₄×D₈	1	1	yes	characteristic
88	C₄	#467	1	1	yes	characteristic
89	C₄	#500	1	1	yes	characteristic
90	C₂	—	2	1	no	—
91	C₂	—	2	1	no	—
92	C₂	#1942	1	1	yes	characteristic
93	C₂	#2221	1	1	yes	characteristic
94	C₂	#2222	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	#466
commutator	C₂×C₈	C₂²×C₄
Frattini, cosocle	C₂²×C₈	C₂³
prime radical	C₂³	#135
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
12. #368	C₄	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#368	2
C₂×#366	2
#2108	2
#1952	2
#1387	2